Walled Klein-4 Brauer algebras

Abstract

The new class of diagram algebras known as walled Klein-$4$ Brauer algebras, denoted by $\overrightarrow{\overrightarrow{D}}_{r,s}(l)$, where $r,s \in \mathds{N}$ and $l \in K$, is an indeterminate, is studied in this paper. The walled klein-$4$ Brauer algebras are explained in terms of generators and relations. The indexing set of the simple modules of the walled Klein-$4$ Brauer algebras was described. We established that walled Klein-$4$ Brauer algebras are iterated inflations of the group algebra of the group $(\mathds{Z}_{2}\times \mathds{Z}_{2})\wr \mathbf{S}_{r} \times (\mathds{Z}_{2}\times \mathds{Z}_{2})\wr \mathbf{S}_{s}$, and we concluded that $\overrightarrow{\overrightarrow{D}}_{r,s}(l)$ is cellular as a consequence.